Question

a figure has the dimensions shown in inches.

1. find the area of the compound figure. leave your answer in factored form.
2. if ( x = 6 ), what is the area of the figure?

Explanation:

Question 1

Step1: Identify the two shapes (triangle and rectangle)

The compound figure consists of a triangle (top) and a rectangle (bottom). For the triangle: base = \( x^2 + 5 \), height = \( 4x^3 \). For the rectangle: length = \( x^2 + 5 \), width = \( 6 \).

Step2: Calculate area of triangle

Area of triangle: \( \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}(x^2 + 5)(4x^3) \)
Simplify: \( 2x^3(x^2 + 5) \)

Step3: Calculate area of rectangle

Area of rectangle: \( \text{length} \times \text{width} = 6(x^2 + 5) \)

Step4: Total area (sum of both areas)

Total area = \( 2x^3(x^2 + 5) + 6(x^2 + 5) \)
Factor out \( (x^2 + 5) \): \( (x^2 + 5)(2x^3 + 6) \)
Factor \( 2x^3 + 6 \) (GCF 2): \( (x^2 + 5) \times 2(x^3 + 3) = 2(x^2 + 5)(x^3 + 3) \)

Step1: Substitute \( x = 6 \) into the factored area formula

From Question 1, area = \( 2(x^2 + 5)(x^3 + 3) \). Substitute \( x = 6 \):

Step2: Calculate \( x^2 + 5 \) when \( x = 6 \)

\( 6^2 + 5 = 36 + 5 = 41 \)

Step3: Calculate \( x^3 + 3 \) when \( x = 6 \)

\( 6^3 + 3 = 216 + 3 = 219 \)

Step4: Compute total area

Total area = \( 2 \times 41 \times 219 \)
First, \( 41 \times 219 = 8979 \), then \( 2 \times 8979 = 17958 \)

Answer:

\( 2(x^2 + 5)(x^3 + 3) \)

Question 2